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The contact integral, which is repeated here, $$\begin{equation} G^{c}\left(\boldsymbol{\varphi},\mathbf{w}\right)=\int\limits _{\Gamma^{\left(1\right)}}t_{N}\delta g\,d\Gamma\,,\label{eq577} \end{equation}$$ will now be discretized using a standard finite element procedure. First it is noted that the integration can be written as a sum over the surface element areas: $$\begin{equation} G^{c}\left(\boldsymbol{\varphi},\mathbf{w}\right)=\sum\limits _{e=1}^{N_{sel}}\int\limits _{\Gamma^{\left(1\right)e}}t_{N}\delta g\,d\Gamma\,,\label{eq578} \end{equation}$$ where $N_{sel}$ is the number of surface elements. The integration can be approximated using a quadrature rule, $$\begin{equation} G^{c}\left(\boldsymbol{\varphi},\mathbf{w}\right)\cong\sum\limits _{e=1}^{N_{sel}}\left\{ \sum\limits _{i=1}^{N_{int}^{e}}w^{i}j\left(\boldsymbol{\xi}_{i}\right)t_{N}\left(\boldsymbol{\xi}_{i}\right)\delta g\left(\boldsymbol{\xi}_{i}\right)\right\} \,,\label{eq579} \end{equation}$$ where $N_{int}^{e}$ are the number of integration points for element $e$ . It is now assumed that the integration points coincide with the element's nodes (e.g. for a quadrilateral surface element we have $\boldsymbol{\xi}_{1}=\left(-1,-1\right)$ , $\boldsymbol{\xi}_{2}=\left(1,-1\right)$ , $\boldsymbol{\xi}_{3}=\left(1,1\right)$ and $\boldsymbol{\xi}_{4}=\left(-1,1\right))$ . With this quadrature rule, we have $$\begin{equation} \begin{aligned}w^{\left(1\right)}\left(\boldsymbol{\xi}_{i}\right) & =\mathbf{c}_{i}^{\left(1\right)}\\ w^{\left(2\right)}\left(\bar{\boldsymbol{\xi}}_{i}\right) & =\sum\limits _{j=1}^{n}N_{j}\left(\bar{\boldsymbol{\xi}}_{i}\right)\mathbf{c}_{j}^{\left(2\right)} \end{aligned} \,,\label{eq580} \end{equation}$$ so that, $$\begin{equation} \delta g\left(\boldsymbol{\xi}_{i}\right)=-\boldsymbol{\nu}\cdot\left(\mathbf{c}_{i}^{\left(1\right)}-\sum\limits _{j=1}^{n}N_{j}^{\left(2\right)}\left(\bar{\boldsymbol{\xi}}_{i}\right)\mathbf{c}_{j}^{\left(2\right)}\right)\,.\label{eq581} \end{equation}$$ If the following vectors are defined, $$\begin{equation} \begin{aligned}\delta\boldsymbol{\Phi}^{T} & =\left[\begin{array}{cccc} \mathbf{c}_{i}^{\left(1\right)} & \mathbf{c}_{1}^{\left(2\right)} & \cdots & \mathbf{c}_{n}^{\left(2\right)}\end{array}\right]\\ \mathbf{N}^{T} & =\left[\begin{array}{cccc} \boldsymbol{\nu} & -\boldsymbol{\nu}N_{1}^{\left(2\right)} & \cdots & -\boldsymbol{\nu}N_{n}^{\left(2\right)}\end{array}\right] \end{aligned} \,,\label{eq582} \end{equation}$$ equation (7.1.4-3) can then be rewritten as follows, $$\begin{equation} G^{c}\left(\boldsymbol{\varphi},\mathbf{w}\right)\cong\sum\limits _{e=1}^{N_{sel}}\left\{ \sum\limits _{i=1}^{N_{int}^{e}}w^{i}j\left(\boldsymbol{\xi}_{i}\right)t_{N}\left(\boldsymbol{\xi}_{i}\right)\delta\boldsymbol{\Phi}^{T}\mathbf{N}^{T}\right\} \,.\label{eq583} \end{equation}$$ The specific form for $t_{N}$ will depend on the method employed for enforcing the contact constraint.